3.6 \(\int \frac{(a+b x^3) (A+B x^3)}{x^3} \, dx\)

Optimal. Leaf size=28 \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

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Rubi [A]  time = 0.0159127, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {448} \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)*(A + B*x^3))/x^3,x]

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right ) \left (A+B x^3\right )}{x^3} \, dx &=\int \left (A b \left (1+\frac{a B}{A b}\right )+\frac{a A}{x^3}+b B x^3\right ) \, dx\\ &=-\frac{a A}{2 x^2}+(A b+a B) x+\frac{1}{4} b B x^4\\ \end{align*}

Mathematica [A]  time = 0.0086834, size = 28, normalized size = 1. \[ x (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{4} b B x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)*(A + B*x^3))/x^3,x]

[Out]

-(a*A)/(2*x^2) + (A*b + a*B)*x + (b*B*x^4)/4

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{bB{x}^{4}}{4}}+Abx+Bax-{\frac{Aa}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)*(B*x^3+A)/x^3,x)

[Out]

1/4*b*B*x^4+A*b*x+B*a*x-1/2*a*A/x^2

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Maxima [A]  time = 1.26629, size = 32, normalized size = 1.14 \begin{align*} \frac{1}{4} \, B b x^{4} +{\left (B a + A b\right )} x - \frac{A a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^3,x, algorithm="maxima")

[Out]

1/4*B*b*x^4 + (B*a + A*b)*x - 1/2*A*a/x^2

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Fricas [A]  time = 1.38094, size = 63, normalized size = 2.25 \begin{align*} \frac{B b x^{6} + 4 \,{\left (B a + A b\right )} x^{3} - 2 \, A a}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^3,x, algorithm="fricas")

[Out]

1/4*(B*b*x^6 + 4*(B*a + A*b)*x^3 - 2*A*a)/x^2

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Sympy [A]  time = 0.258407, size = 24, normalized size = 0.86 \begin{align*} - \frac{A a}{2 x^{2}} + \frac{B b x^{4}}{4} + x \left (A b + B a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)*(B*x**3+A)/x**3,x)

[Out]

-A*a/(2*x**2) + B*b*x**4/4 + x*(A*b + B*a)

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Giac [A]  time = 1.15505, size = 31, normalized size = 1.11 \begin{align*} \frac{1}{4} \, B b x^{4} + B a x + A b x - \frac{A a}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)*(B*x^3+A)/x^3,x, algorithm="giac")

[Out]

1/4*B*b*x^4 + B*a*x + A*b*x - 1/2*A*a/x^2